Editing Inverse-square law

{{short description|Physical law}}
{{Use dmy dates|date=June 2020}}
[[Image:Inverse square law.svg|thumb|420px|S represents the light source, while r represents the measured points. The lines represent the [[flux]] emanating from the sources and fluxes.
 
The total number of [[flux line]]s depends on the strength of the light source and is constant with increasing distance, where a greater density of flux lines (lines per unit area) means a stronger energy field. The density of flux lines is inversely proportional to the square of the distance from the source because the surface area of a sphere increases with the square of the radius. Thus the field intensity is inversely proportional to the square of the distance from the source.]]

In [[science]], an '''inverse-square law''' is any [[scientific law]] stating that the observed "intensity" of a specified [[physical quantity]] is [[Proportionality (mathematics)#Inverse proportionality|inversely proportional]] to the [[square (algebra)|square]] of the [[distance]] from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.

[[Radar]] energy expands during both the signal transmission and the [[reflection (physics)|reflected]] return, so the inverse square for both paths means that the radar will receive energy according to the inverse [[fourth power]] of the range.

To prevent dilution of [[energy]] while propagating a signal, certain methods can be used such as a [[waveguide]], which acts like a canal does for water, or how a gun barrel restricts hot gas expansion to one [[dimension]] in order to prevent loss of energy transfer to a [[bullet]].

== Formula ==
In mathematical notation the inverse square law can be expressed as an intensity (I) varying as a function of distance (d) from some centre. The intensity is proportional (see [[∝]]) to the reciprocal of the square of the distance thus: 
<math display="block">\text{intensity} \ \propto \ \frac{1}{\text{distance}^2} \, </math>

It can also be mathematically expressed as :
<math display="block">\frac{\text{intensity}_1}{\text{intensity}_2} = \frac{\text{distance}_2^2}{\text{distance}_1^2}</math>

or as the formulation of a constant quantity:   
<math display="block">\text{intensity}_1 \times \text{distance}_1^2 = \text{intensity}_2 \times \text{distance}_2^2</math>

The [[divergence]] of a [[vector field]] which is the resultant of radial inverse-square law fields with respect to one or more sources is proportional to the strength of the local sources, and hence zero outside sources. [[Newton's law of universal gravitation]] follows an inverse-square law, as do the effects of [[electricity|electric]], [[light]], [[sound]], and [[radiation]] phenomena.

== Justification ==
The inverse-square law generally applies when some force, energy, or [[flux|other conserved quantity]] is evenly radiated outward from a [[point source]] in [[three-dimensional space]].  Since the [[surface area]] of a [[sphere]] (which is&nbsp;4π''r''<sup>2</sup>) is proportional to the square of the radius, as the [[flux density|emitted radiation]] gets farther from the source, it is spread out over an area that is increasing in proportion to the square of the distance from the source. Hence, the intensity of radiation passing through any unit area (directly facing the point source) is inversely proportional to the square of the distance from the point source. [[Gauss's law for gravity]] is similarly applicable, and can be used with any physical quantity that acts in accordance with the inverse-square relationship.

==Occurrences==

===Gravitation===
[[Gravity|Gravitation]] is the attraction between objects that have mass. Newton's law states:

{{quote|The gravitational attraction force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of their separation distance. The force is always attractive and acts along the line joining them.<ref name=Newton1>Proposition 75, Theorem 35: p. 956 – I.Bernard Cohen and Anne Whitman, translators: [[Isaac Newton]], ''The Principia'': [[Mathematical Principles of Natural Philosophy]]. Preceded by ''A Guide to Newton's Principia'', by I.Bernard Cohen. University of California Press 1999 {{ISBN|0-520-08816-6}} {{ISBN|0-520-08817-4}}</ref>}}

<math display="block">F=G\frac{m_1 m_2}{r^2}</math>

If the distribution of matter in each body is spherically symmetric, then the objects can be treated as point masses without approximation, as shown in the [[shell theorem]].  Otherwise, if we want to calculate the attraction between massive bodies, we need to add all the point-point attraction forces vectorially and the net attraction might not be exact inverse square. However, if the separation between the massive bodies is much larger compared to their sizes, then to a good approximation, it is reasonable to treat the masses as a point mass located at the object's [[center of mass]] while calculating the gravitational force.

As the law of gravitation, this [[Law of universal gravitation|law]] was suggested in 1645 by [[Ismaël Bullialdus]]. But Bullialdus did not accept [[Kepler's laws of planetary motion|Kepler's second and third laws]], nor did he appreciate [[Christiaan Huygens]]'s solution for circular motion (motion in a straight line pulled aside by the central force).  Indeed, Bullialdus maintained the sun's force was attractive at aphelion and repulsive at perihelion. [[Robert Hooke]] and [[Giovanni Alfonso Borelli]] both expounded gravitation in 1666 as an attractive force.<ref>Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses. See page 239 in: {{Cite book |title=General history of astronomy |date=2009 |publisher=[[Cambridge University Press]] |isbn=978-0-521-54205-0 |editor-last=Taton |editor-first=René |editor-link=René Taton |edition=1. |volume=2 Pt. A: Planetary astronomy from the Renaissance to the rise of astrophysics Tycho Brahe to Newton |location=Cambridge |pages=233–274 |chapter=The Newtonian achievement in astronomy |editor-last2=Wilson |editor-first2=Curtis |editor-last3=Hoskin |editor-first3=Michael A.}}</ref> Hooke's lecture "On gravity" was at the Royal Society, in London, on 21 March.<ref>{{Cite book |last=Birch |first=Thomas |author-link=Thomas Birch |url=https://books.google.com/books?id=lWEVAAAAQAAJ |title=The History of the Royal Society of London |date=1756 |volume=2 |pages=68–73}}; see especially pages 70–72.</ref> Borelli's "Theory of the Planets" was published later in 1666.<ref>{{Cite book |last=Borelli |first=Giovanni Alfonso |author-link=Giovanni Alfonso Borelli |url=https://books.google.com/books?id=YZk_AAAAcAAJ |title=Theoricae mediceorum planetarum ex causis physicis deductae |date=1666 |publisher=Ex typographia S.M.D. |bibcode=1666tmpe.book.....B |language=la |trans-title=Theory [of the motion] of the Medicean planets [i.e., moons of Jupiter] deduced from physical causes}}</ref>  Hooke's 1670 Gresham lecture explained that gravitation applied to "all celestiall bodys" and added the principles that the gravitating power decreases with distance and that in the absence of any such power bodies move in straight lines. By 1679, Hooke thought gravitation had inverse square dependence and communicated this in a letter to [[Isaac Newton]]:<ref name="Hooke1680">{{Cite journal |last=Koyré |first=Alexandre |author-link=Alexandre Koyré |year=1952 |title=An Unpublished Letter of Robert Hooke to Isaac Newton |journal=[[Isis (journal)|Isis]] |volume=43 |issue=4 |pages=312–337 |doi=10.1086/348155 |jstor=227384 |pmid=13010921 }}</ref> 
''my supposition is that the attraction always is in duplicate proportion to the distance from the center reciprocall''.<ref>Hooke's letter to Newton of 6 January 1680 (Koyré 1952:332).</ref>

Hooke remained bitter about Newton claiming the invention of this principle, even though Newton's 1686 ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' acknowledged that Hooke, along with Wren and Halley, had separately appreciated the inverse square law in the [[Solar System]],<ref>Newton acknowledged Wren, Hooke and Halley in this connection in the Scholium to Proposition 4 in Book 1 (in all editions): See for example: {{Cite book |last=Newton |first=Isaac |author-link=Isaac Newton |url=https://books.google.com/books?id=Tm0FAAAAQAAJ |title=The Mathematical Principles of Natural Philosophy |date=1729 |publisher=B. Motte |page=66}}</ref> as well as giving some credit to Bullialdus.<ref>In a letter to Edmund Halley dated 20 June 1686, Newton wrote:  "Bullialdus wrote that all force respecting ye Sun as its center & depending on matter must be reciprocally in a duplicate ratio of ye distance from ye center." See: {{Cite book |last1=Cohen |first1=I. Bernard |author-link=I. Bernard Cohen |url=https://books.google.com/books?id=3wIzvqzfUXkC |title=The Cambridge Companion to Newton |last2=Smith |first2=George E. |author-link2=George E. Smith |date=2002 |publisher=Cambridge University Press |isbn=978-0-521-65696-2 |pages=204}}</ref>

===Electrostatics===
{{Main|Electrostatics}}
The force of attraction or repulsion between two electrically charged particles, in addition to being directly proportional to the product of the electric charges, is inversely proportional to the square of the distance between them; this is known as [[Coulomb's law]]. The deviation of the exponent from 2 is less than one part in 10<sup>15</sup>.<ref>{{Cite journal |last1=Williams |first1=E. R. |last2=Faller |first2=J. E. |last3=Hill |first3=H. A. |year=1971 |title=New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass |journal=[[Physical Review Letters]] |language=en |volume=26 |issue=12 |pages=721–724 |bibcode=1971PhRvL..26..721W |doi=10.1103/PhysRevLett.26.721 }}</ref>

<math display="block">F=k_\text{e}\frac{q_1 q_2}{r^2}</math>

===Light and other electromagnetic radiation===
The [[intensity (physics)|intensity]] (or [[illuminance]] or [[irradiance]]) of [[light]] or other linear waves radiating from a [[point source]] (energy per unit of area perpendicular to the source) is inversely proportional to the square of the distance from the source, so an object (of the same size) twice as far away receives only one-quarter the [[energy]] (in the same time period).

More generally, the irradiance, ''i.e.,'' the intensity (or [[power (physics)|power]] per unit area in the direction of [[wave propagation|propagation]]), of a [[sphere|spherical]] [[wavefront]] varies inversely with the square of the distance from the source (assuming there are no losses caused by [[absorption (optics)|absorption]] or [[scattering]]).

For example, the intensity of radiation from the [[Sun]] is 9126 [[watt]]s per square meter at the distance of [[Mercury (planet)|Mercury]] (0.387 [[Astronomical unit|AU]]); but only 1367 watts per square meter at the distance of [[Earth]] (1 AU)—an approximate threefold increase in distance results in an approximate ninefold decrease in intensity of radiation.

For non-[[isotropic radiator]]s such as [[parabolic antenna]]s, headlights, and [[laser]]s, the effective origin is located far behind the beam aperture. If you are close to the origin, you don't have to go far to double the radius, so the signal drops quickly. When you are far from the origin and still have a strong signal, like with a laser, you have to travel very far to double the radius and reduce the signal. This means you have a stronger signal or have [[antenna gain]] in the direction of the narrow beam relative to a wide beam in all directions of an [[Isotropic radiator|isotropic antenna]].

In [[photography]] and [[stage lighting]], the inverse-square law is used to determine the “fall off” or the difference in illumination on a subject as it moves closer to or further from the light source. For quick approximations, it is enough to remember that doubling the distance reduces illumination to one quarter;<ref>{{Cite book |last=Millerson |first=Gerald |url=https://books.google.com/books?id=Kf6XAAAAQBAJ |title=Lighting for TV and Film |date=1999 |publisher=CRC Press |isbn=978-1-136-05522-5 |pages=27}}</ref> or similarly, to halve the illumination increase the distance by a factor of 1.4 (the [[square root of 2]]), and to double illumination, reduce the distance to 0.7 (square root of 1/2). When the illuminant is not a point source, the inverse square rule is often still a useful approximation; when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1%.<ref>{{Cite book |last=Ryder |first=Alexander D. |url=https://cgvr.informatik.uni-bremen.de/teaching/cg_literatur/ILT-Light-Measurement-Handbook.pdf |title=The Light Measurement Handbook |date=1997 |publisher=International Light |isbn=978-0-96-583569-5 |pages=26}}</ref>

The fractional reduction in electromagnetic [[fluence]] (Φ) for indirectly ionizing radiation with increasing distance from a point source can be calculated using the inverse-square law. Since emissions from a point source have radial directions, they intercept at a perpendicular incidence. The area of such a shell is 4π''r'' <sup>2</sup> where ''r'' is the radial distance from the center. The law is particularly important in diagnostic [[radiography]] and [[radiotherapy]] treatment planning, though this proportionality does not hold in practical situations unless source dimensions are much smaller than the distance.  As stated in [[Fourier theory]] of heat “as the point source is magnification by distances, its radiation is dilute proportional to the sin of the angle, of the increasing circumference arc from the point of origin”.

====Example====
Let ''P''&nbsp; be the total power radiated from a point source (for example, an omnidirectional [[isotropic radiator]]). At large distances from the source (compared to the size of the source), this power is distributed over larger and larger spherical surfaces as the distance from the source increases.  Since the surface area of a sphere of radius ''r'' is ''A''&nbsp;=&nbsp;4''πr''<sup>&nbsp;2</sup>, the [[intensity (physics)|intensity]] ''I'' (power per unit area) of radiation at distance ''r'' is
<math display="block">
I = \frac P A = \frac P {4 \pi r^2}. \,
</math>

The energy or intensity decreases (divided by&nbsp;4) as the distance ''r'' is doubled; if measured in [[Decibel|dB]] would decrease by 6.02&nbsp;dB per doubling of distance. When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference value.

===Sound in a gas===

In [[acoustics]], the [[sound pressure]] of a [[sphere|spherical]] [[wavefront]] radiating from a point source decreases by 50% as the distance ''r'' is doubled; measured in [[Decibel|dB]], the decrease is still 6.02&nbsp;dB, since dB represents an intensity ratio. The pressure ratio (as opposed to power ratio) is not inverse-square, but is inverse-proportional (inverse distance law):

<math display="block"> p \ \propto \ \frac{1}{r} \,
 </math>

The same is true for the component of [[particle velocity]] <math> v \,</math> that is [[Phase (waves)#In-phase and quadrature (I&Q) components|in-phase]] with the instantaneous sound pressure <math>p \,</math>:

<math display="block"> v \ \propto \frac{1}{r} \ \, </math>

In the [[Near and far field|near field]] is a [[quadrature phase|quadrature component]] of the particle velocity that is 90° out of phase with the sound pressure and does not contribute to the time-averaged energy or the intensity of the sound.  The [[sound intensity]] is the product of the [[root mean square|RMS]] sound pressure and the ''in-phase'' component of the RMS particle velocity, both of which are inverse-proportional. Accordingly, the intensity follows an inverse-square behaviour:

<math display="block"> I \ = \ p v \ \propto \ \frac{1}{r^2}. \, </math>

==Field theory interpretation==
For an [[irrotational vector field]] in three-dimensional space, the inverse-square law corresponds to the property that the [[divergence]] is zero outside the source. This can be generalized to higher dimensions. Generally, for an irrotational vector field in ''n''-dimensional [[Euclidean space]], the intensity "I" of the vector field falls off with the distance "r" following the inverse (''n''&nbsp;−&nbsp;1)<sup>th</sup> power law

<math display="block">I\propto \frac 1 {r^{n-1}},</math>

given that the space outside the source is divergence free. {{Citation needed|date=March 2011}}

== Non-Euclidean implications ==

The inverse-square law, fundamental in [[Euclidean geometry|Euclidean]] spaces, also applies to [[Non-Euclidean geometry|non-Euclidean geometries]], including [[Hyperbolic geometry|hyperbolic space]]. The curvature present in these spaces alters physical laws, influencing a variety of fields such as [[cosmology]], [[general relativity]], and [[string theory]].<ref>{{Cite journal |last=Barrow |first=John D |date=2020 |title=Non-Euclidean Newtonian cosmology |journal=[[Classical and Quantum Gravity]] |volume=37 |issue=12 |pages=125007 |arxiv=2002.10155 |bibcode=2020CQGra..37l5007B |doi=10.1088/1361-6382/ab8437 }}</ref>

[[John D. Barrow]], in his 2020 paper "Non-Euclidean Newtonian Cosmology," expands on the behavior of force (F) and potential (Φ) within hyperbolic 3-space (H3). He explains that F and Φ obey the relationships F ∝ 1 / R² sinh²(r/R) and Φ ∝ coth(r/R), where R represents the curvature radius and r represents the distance from the focal point.

The concept of spatial dimensionality, first proposed by Immanuel Kant, remains a topic of debate concerning the inverse-square law.<ref>{{Cite journal |last1=Gatzia |first1=Dimitria Electra |last2=Ramsier |first2=Rex D. |date=2021 |title=Dimensionality, symmetry and the Inverse Square Law |journal=Notes and Records: The Royal Society Journal of the History of Science |volume=75 |issue=3 |pages=333–348 |doi=10.1098/rsnr.2019.0044 }}</ref> Dimitria Electra Gatzia and Rex D. Ramsier, in their 2021 paper, contend that the inverse-square law is more closely related to force distribution symmetry than to the dimensionality of space.

In the context of non-Euclidean geometries and general relativity, deviations from the inverse-square law do not arise from the law itself but rather from the assumption that the force between two bodies is instantaneous, which contradicts [[special relativity]]. General relativity reinterprets gravity as the curvature of spacetime, leading particles to move along geodesics in this curved spacetime.<ref>{{Cite web |last=Guth |first=Alan |date=2018 |title=Introduction to Non-Euclidean General Relativity |url=https://web.mit.edu/8.286/www/lecn18/ln05-euf18.pdf |access-date=2023-07-30 |publisher=[[MIT OpenCourseWare]]}}</ref>

==History==

[[John Dumbleton]] of the 14th-century [[Oxford Calculators]], was one of the first to express functional relationships in graphical form. He gave a proof of the [[mean speed theorem]] stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint" and used this method to study the quantitative decrease in intensity of illumination in his ''Summa logicæ et philosophiæ naturalis'' (ca. 1349), stating that it was not linearly proportional to the distance, but was unable to expose the Inverse-square law.<ref>{{Cite book |last=Freely |first=John |author-link=John Freely |title=Before Galileo: The Birth of Modern Science in Medieval Europe |date=2012 |publisher=Overlook Duckworth |isbn=978-0-71-564536-9}}</ref>
[[File:Kepler_1910.jpg|alt=Kepler 1910|thumb|199x199px|German astronomer [[Johannes Kepler]] discussed the inverse-square law and how it affects the intensity of light.]]
In proposition 9 of Book 1 in his book ''Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur'' (1604), the astronomer [[Johannes Kepler]] argued that the spreading of light from a point source obeys an inverse square law:<ref>{{Cite book |last=Kepler |first=Johannes |author-link=Johannes Kepler |url=https://books.google.com/books?id=-OB9O7FowP4C |title=Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur |date=1604 |publisher=Apud Claudium Marnium & haeredes Ioannis Aubrii |page=10 |language=la}}</ref><ref>Translation of the Latin quote from Kepler's ''Ad Vitellionem paralipomena'' is from: {{Cite journal |last1=Gal |first1=Ofer |last2=Chen-Morris |first2=Raz |date=2005 |title=The Archaeology of the Inverse Square Law: (1) Metaphysical Images and Mathematical Practices |journal=History of Science |volume=43 |issue=4 |pages=391–414 |doi=10.1177/007327530504300402 |bibcode=2005HisSc..43..391G |bibcode-access=free}} ; see especially p. 397.</ref>

{{verse translation|lang=la|
Sicut se habent spharicae superificies, quibus origo lucis pro centro est, amplior ad angustiorem:  ita se habet fortitudo seu densitas lucis radiorum in angustiori, ad illamin in laxiori sphaerica, hoc est, conversim.  Nam per 6. 7. tantundem lucis est in angustiori sphaerica superficie, quantum in fusiore, tanto ergo illie stipatior & densior quam hic.
|
Just as [the ratio of] spherical surfaces, for which the source of light is the center, [is] from the wider to the narrower, so the density or fortitude of the rays of light in the narrower [space], towards the more spacious spherical surfaces, that is, inversely.  For according to [propositions] 6 & 7, there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there.
}}


In 1645, in his book ''Astronomia Philolaica'' ..., the French astronomer [[Ismaël Bullialdus]] (1605–1694) refuted Johannes Kepler's suggestion that "gravity"<ref>Note:  Both Kepler and William Gilbert had nearly anticipated the modern conception of gravity, lacking only the inverse-square law in their description of "gravitas". On page 4 of chapter 1, Introductio, of ''Astronomia Nova'', Kepler sets out his description as follows:

''"The true theory of gravity is founded on the following axioms:''

''Every corporeal substance, so far forth as it is corporeal, has a natural fitness for resting in every place where it may be situated by itself beyond the sphere of influence of a body cognate with it.''

'''''Gravity is a mutual affection between cognate bodies towards union or conjunction (similar in kind to the magnetic virtue), so that the earth attracts a stone much rather than the stone seeks the earth.'''''

...

''If two stones were placed in any part of the world near each other, and beyond the sphere of influence of a third cognate body, these stones, like two magnetic needles, would come together in the intermediate point, '''each approaching the other by a space proportional to the comparative mass of the other'''.''

''If the moon and earth were not retained in their orbits by their animate force or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth through the other fifty-three parts, and they would there meet, assuming, however, that the substance of both is of the same density."''

Notice that in saying "''the earth attracts a stone much rather than the stone seeks the earth"'' Kepler is breaking away from the Aristotelian tradition that objects ''seek'' to be in their natural place, that a stone ''seeks'' to be with the earth.</ref> weakens as the inverse of the distance; instead, Bullialdus argued, "gravity" weakens as the inverse square of the distance:<ref>{{Cite book |last=Boulliau |first=Ismael |author-link=Ismaël Bullialdus |url=https://books.google.com/books?id=BkZZAAAAcAAJ |title=Astronomia Philolaica |date=1645 |publisher=Simeon Piget |page=23 |language=la |bibcode=1645ibap.book.....B |doi=10.3931/e-rara-549 |bibcode-access=free}}</ref><ref>Translation of the Latin quote from Bullialdus' 'Astronomia Philolaica' … is from: {{Cite web |last1=O'Connor |first1=John J. |last2=Robertson |first2=Edmund F. |date=2006 |title=Ismael Boulliau |url=https://mathshistory.st-andrews.ac.uk/Biographies/Boulliau/ |website=The MacTutor History of Mathematics Archive |publisher=[[University of Saint Andrews]]}}</ref>

{{verse translation|lang=la|
Virtus autem illa, qua Sol prehendit seu harpagat planetas, corporalis quae ipsi pro manibus est, lineis rectis in omnem mundi amplitudinem emissa quasi species solis cum illius corpore rotatur:  cum ergo sit corporalis imminuitur, & extenuatur in maiori spatio & intervallo, ratio autem huius imminutionis eadem est, ac luminus, in ratione nempe dupla intervallorum, sed eversa.
|
As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances [that is, 1/d²].
}}

In England, the Anglican bishop [[Seth Ward (bishop of Salisbury)|Seth Ward]] (1617–1689) publicized the ideas of Bullialdus in his critique ''In Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis'' (1653) and publicized the planetary astronomy of Kepler in his book ''Astronomia geometrica'' (1656).

In 1663–1664, the English scientist [[Robert Hooke]] was writing his book ''Micrographia'' (1666) in which he discussed, among other things, the relation between the height of the atmosphere and the barometric pressure at the surface.  Since the atmosphere surrounds the Earth, which itself is a sphere, the volume of atmosphere bearing on any unit area of the Earth's surface is a truncated cone (which extends from the Earth's center to the vacuum of space; obviously only the section of the cone from the Earth's surface to space bears on the Earth's surface).  Although the volume of a cone is proportional to the cube of its height, Hooke argued that the air's pressure at the Earth's surface is instead proportional to the height of the atmosphere because gravity diminishes with altitude.  Although Hooke did not explicitly state so, the relation that he proposed would be true only if gravity decreases as the inverse square of the distance from the Earth's center.<ref>{{harvnb|Gal|Chen-Morris|2005|pp=391–392}}</ref><ref>{{Cite book |last=Hooke |first=Robert |author-link= |url=https://books.google.com/books?id=LsbBada4VVYC |title=Micrographia |date= |publisher=Science Heritage |isbn=978-0-940095-07-6 |pages=227 |quote=I say a ''Cylinder'', not a piece of a ''Cone'', because, as I may elsewhere shew in the Explication of Gravity, that ''triplicate'' proportion of the shels of a Sphere, to their respective diameters, I suppose to be removed in this case by the decrease of the power of Gravity.}}</ref>

==See also==
* [[Flux]]
* [[Antenna (radio)]]
* [[Gauss's law]]
* [[Kepler's laws of planetary motion]]
* [[Kepler problem]]
* [[Telecommunications]], particularly:
** [[William Thomson, 1st Baron Kelvin#Calculations on data rate|William Thomson, 1st Baron Kelvin]]
** [[List of ad hoc routing protocols#Power-aware routing protocols|Power-aware routing protocols]]
* [[Inverse proportion#Inverse proportionality|Inverse proportionality]]
* [[Multiplicative inverse]]
* [[Distance decay]]
* [[Fermi paradox]]
* [[Square–cube law]]
* [[Principle of similitude]]

==References==
{{FS1037C}}
{{reflist}}

==External links==
* [http://www.sengpielaudio.com/calculator-distance.htm Damping of sound level with distance]
* [http://www.sengpielaudio.com/calculator-distancelaw.htm Sound pressure p and the inverse distance law 1/r]

{{DEFAULTSORT:Inverse-Square Law}}
[[Category:Philosophy of physics]]
[[Category:Scientific method]]
[[Category:Scientific laws]]